阶数为n的幻方是在一个正方形中排列n 2个数字(通常是不同的整数)的排列，以使所有行，所有列和对角线中的n个数字求和为同一常数。幻方包含从1到n 2的整数。
每行，每一列和对角线中的常数总和称为魔术常数或魔术总和M。正常魔术方阵的魔术常数仅取决于n，并且具有以下值：
M = n(n 2 +1)/ 2
For normal magic squares of order n = 3, 4, 5, ...,
the magic constants are: 15, 34, 65, 111, 175, 260, ... 在本文中，我们将讨论如何以编程方式生成大小为n的幻方。在继续之前，请考虑以下示例：
Magic Square of size 3
-----------------------
2 7 6
9 5 1
4 3 8
Sum in each row & each column = 3*(32+1)/2 = 15
Magic Square of size 5
----------------------
9 3 22 16 15
2 21 20 14 8
25 19 13 7 1
18 12 6 5 24
11 10 4 23 17
Sum in each row & each column = 5*(52+1)/2 = 65
Magic Square of size 7
----------------------
20 12 4 45 37 29 28
11 3 44 36 35 27 19
2 43 42 34 26 18 10
49 41 33 25 17 9 1
40 32 24 16 8 7 48
31 23 15 14 6 47 39
22 21 13 5 46 38 30
Sum in each row & each column = 7*(72+1)/2 = 175您是否找到了存储数字的任何模式?
在任何幻方中，第一个数字(即1)存储在位置(n / 2，n-1)。将此位置设为(i，j)。下一个数字存储在位置(i-1，j + 1)，在这里我们可以将每一行和每一列视为圆形数组，即它们环绕。
符合以下三个条件：
1.下一个数字的位置是通过将前一个数字的行号减1，然后再将前一个数字的列号增1来计算的。在任何时候，如果计算出的行位置变为-1，它将环绕到n-1。同样，如果计算出的列位置变为n，则它将环绕为0。
2.如果幻方在计算位置处已经包含数字，则计算列位置将减少2，计算行位置将增加1。
3.如果计算出的行位置为-1且计算出的列位置为n，则新位置将为：(0，n-2)。
Example:
Magic Square of size 3
----------------------
2 7 6
9 5 1
4 3 8
Steps:
1. position of number 1 = (3/2, 3-1) = (1, 2)
2. position of number 2 = (1-1, 2+1) = (0, 0)
3. position of number 3 = (0-1, 0+1) = (3-1, 1) = (2, 1)
4. position of number 4 = (2-1, 1+1) = (1, 2)
Since, at this position, 1 is there. So, apply condition 2.
new position=(1+1,2-2)=(2,0)
5. position of number 5=(2-1,0+1)=(1,1)
6. position of number 6=(1-1,1+1)=(0,2)
7. position of number 7 = (0-1, 2+1) = (-1,3) // this is tricky, see condition 3
new position = (0, 3-2) = (0,1)
8. position of number 8=(0-1,1+1)=(-1,2)=(2,2) //wrap around
9. position of number 9=(2-1,2+1)=(1,3)=(1,0) //wrap around基于上述方法，以下是工作代码：
C++
// C++ program to generate odd sized magic squares
#include
using namespace std;
// A function to generate odd sized magic squares
void generateSquare(int n)
{
int magicSquare[n][n];
// set all slots as 0
memset(magicSquare, 0, sizeof(magicSquare));
// Initialize position for 1
int i = n / 2;
int j = n - 1;
// One by one put all values in magic square
for (int num = 1; num <= n * n;) {
if (i == -1 && j == n) // 3rd condition
{
j = n - 2;
i = 0;
}
else {
// 1st condition helper if next number
// goes to out of square's right side
if (j == n)
j = 0;
// 1st condition helper if next number
// is goes to out of square's upper side
if (i < 0)
i = n - 1;
}
if (magicSquare[i][j]) // 2nd condition
{
j -= 2;
i++;
continue;
}
else
magicSquare[i][j] = num++; // set number
j++;
i--; // 1st condition
}
// Print magic square
cout << "The Magic Square for n=" << n
<< ":\nSum of "
"each row or column "
<< n * (n * n + 1) / 2 << ":\n\n";
for (i = 0; i < n; i++) {
for (j = 0; j < n; j++)
// setw(7) is used so that the matrix gets
// printed in a proper square fashion.
cout << setw(4) << magicSquare[i][j] << " ";
cout << endl;
}
}
// Driver code
int main()
{
// Works only when n is odd
int n = 7;
generateSquare(n);
return 0;
}
// This code is contributed by rathbhupendra C
// C program to generate odd sized magic squares
#include
#include
// A function to generate odd sized magic squares
void generateSquare(int n)
{
int magicSquare[n][n];
// set all slots as 0
memset(magicSquare, 0, sizeof(magicSquare));
// Initialize position for 1
int i = n / 2;
int j = n - 1;
// One by one put all values in magic square
for (int num = 1; num <= n * n;) {
if (i == -1 && j == n) // 3rd condition
{
j = n - 2;
i = 0;
}
else {
// 1st condition helper if next number
// goes to out of square's right side
if (j == n)
j = 0;
// 1st condition helper if next number
// is goes to out of square's upper side
if (i < 0)
i = n - 1;
}
if (magicSquare[i][j]) // 2nd condition
{
j -= 2;
i++;
continue;
}
else
magicSquare[i][j] = num++; // set number
j++;
i--; // 1st condition
}
// Print magic square
printf("The Magic Square for n=%d:\nSum of "
"each row or column %d:\n\n",
n, n * (n * n + 1) / 2);
for (i = 0; i < n; i++) {
for (j = 0; j < n; j++)
printf("%3d ", magicSquare[i][j]);
printf("\n");
}
}
// Driver program to test above function
int main()
{
int n = 7; // Works only when n is odd
generateSquare(n);
return 0;
} Java
// Java program to generate odd sized magic squares
import java.io.*;
class GFG {
// Function to generate odd sized magic squares
static void generateSquare(int n)
{
int[][] magicSquare = new int[n][n];
// Initialize position for 1
int i = n / 2;
int j = n - 1;
// One by one put all values in magic square
for (int num = 1; num <= n * n;) {
if (i == -1 && j == n) // 3rd condition
{
j = n - 2;
i = 0;
}
else {
// 1st condition helper if next number
// goes to out of square's right side
if (j == n)
j = 0;
// 1st condition helper if next number is
// goes to out of square's upper side
if (i < 0)
i = n - 1;
}
// 2nd condition
if (magicSquare[i][j] != 0) {
j -= 2;
i++;
continue;
}
else
// set number
magicSquare[i][j] = num++;
// 1st condition
j++;
i--;
}
// print magic square
System.out.println("The Magic Square for " + n
+ ":");
System.out.println("Sum of each row or column "
+ n * (n * n + 1) / 2 + ":");
for (i = 0; i < n; i++) {
for (j = 0; j < n; j++)
System.out.print(magicSquare[i][j] + " ");
System.out.println();
}
}
// driver program
public static void main(String[] args)
{
// Works only when n is odd
int n = 7;
generateSquare(n);
}
}
// Contributed by Pramod KumarPython
# Python program to generate
# odd sized magic squares
# A function to generate odd
# sized magic squares
def generateSquare(n):
# 2-D array with all
# slots set to 0
magicSquare = [[0 for x in range(n)]
for y in range(n)]
# initialize position of 1
i = n / 2
j = n - 1
# Fill the magic square
# by placing values
num = 1
while num <= (n * n):
if i == -1 and j == n: # 3rd condition
j = n - 2
i = 0
else:
# next number goes out of
# right side of square
if j == n:
j = 0
# next number goes
# out of upper side
if i < 0:
i = n - 1
if magicSquare[int(i)][int(j)]: # 2nd condition
j = j - 2
i = i + 1
continue
else:
magicSquare[int(i)][int(j)] = num
num = num + 1
j = j + 1
i = i - 1 # 1st condition
# Printing magic square
print("Magic Squre for n =", n)
print("Sum of each row or column",
n * (n * n + 1) / 2, "\n")
for i in range(0, n):
for j in range(0, n):
print('%2d ' % (magicSquare[i][j]),
end='')
# To display output
# in matrix form
if j == n - 1:
print()
# Driver Code
# Works only when n is odd
n = 7
generateSquare(n)
# This code is contributed
# by Harshit AgrawalC#
// C# program to generate odd sized magic squares
using System;
class GFG {
// Function to generate odd sized magic squares
static void generateSquare(int n)
{
int[, ] magicSquare = new int[n, n];
// Initialize position for 1
int i = n / 2;
int j = n - 1;
// One by one put all values in magic square
for (int num = 1; num <= n * n;) {
if (i == -1 && j == n) // 3rd condition
{
j = n - 2;
i = 0;
}
else {
// 1st condition helper if next number
// goes to out of square's right side
if (j == n)
j = 0;
// 1st condition helper if next number is
// goes to out of square's upper side
if (i < 0)
i = n - 1;
}
// 2nd condition
if (magicSquare[i, j] != 0) {
j -= 2;
i++;
continue;
}
else
// set number
magicSquare[i, j] = num++;
// 1st condition
j++;
i--;
}
// print magic square
Console.WriteLine("The Magic Square for " + n
+ ":");
Console.WriteLine("Sum of each row or column "
+ n * (n * n + 1) / 2 + ":");
for (i = 0; i < n; i++) {
for (j = 0; j < n; j++)
Console.Write(magicSquare[i, j] + " ");
Console.WriteLine();
}
}
// driver program
public static void Main()
{
// Works only when n is odd
int n = 7;
generateSquare(n);
}
}
// This code is contributed by Sam007.PHP
Javascript
输出
The Magic Square for n=7:
Sum of each row or column 175:
20 12 4 45 37 29 28
11 3 44 36 35 27 19
2 43 42 34 26 18 10
49 41 33 25 17 9 1
40 32 24 16 8 7 48
31 23 15 14 6 47 39
22 21 13 5 46 38 30